There have been some previous works on the gradient estimate for complex Monge-Amp\(\grave{\text{e}}\) re. In [1]}, the author showed how to get \(C^1\)  estimate directly from \(C^0\)  estimate, using a maximum principle argument, but this estimate will require a Lipschitz bound on the right hand side. This estimate has been improved in [2]}, in which the assumption is weakened to \(\nabla F\in L^p\)  with \(p>2n\)  and the \(C^{1,1}\)  bound is obtained.
